Theory of fractional Hermite polynomials and related families

Characterize the properties of the fractional Hermite polynomials introduced in Example 7.3, including formulas and orthogonality relations, and extend the analysis to fractional analogues of Legendre and Laguerre polynomials.

Background

Example 7.3 defines fractional Hermite polynomials arising from an L-fractional Hermite equation. Understanding their structure and properties would parallel classical orthogonal polynomial theory in the fractional setting.